The Limit Superior and the Limit Inferior of a Bounded Sequence of Real Numbers

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $(a_{n})$ be a bounded sequence in $\mathbb{R}$, and for $k\in\mathbb{N}$ let $\overline{a}_{k}=\sup\{a_{m}:m\in\mathbb{N},\ m\ge k\}$ and $\underline{a}_{k}=\inf\{a_{m}:m\in\mathbb{N},\ m\ge k\}$ be its tail supremum and tail infimum, which exist by Tail Suprema and Tail Infima of a Bounded Sequence of Real Numbers §tails.

¶ The limit superior of $(a_{n})$, written $\limsup_{n\to\infty}a_{n}$, is the limit $\lim_{k\to\infty}\overline{a}_{k}$ of the sequence $(\overline{a}_{k})_{k\in\mathbb{N}}$, which is convergent by Tail Suprema and Tail Infima of a Bounded Sequence of Real Numbers §limits.

¶ The limit inferior of $(a_{n})$, written $\liminf_{n\to\infty}a_{n}$, is the limit $\lim_{k\to\infty}\underline{a}_{k}$ of the sequence $(\underline{a}_{k})_{k\in\mathbb{N}}$, which is convergent by Tail Suprema and Tail Infima of a Bounded Sequence of Real Numbers §limits.